In this paper, we study a stochastic LQ problem on infinite time horizon with regime switching and random coefficients, where the control variable has to be constrained in a cone. And the control weighting matrix in the cost functional is allowed to be possibly singular. The state process is shown to satisfy both the \(L^2\) -stable condition (see e.g. Definition 2.1 in [1]}) and the mean-square stabilizing condition (see e.g. Definition 2.1 in [2]} or [3]}). To solve the control problem, we introduce two systems of BSDEs termed extended stochastic Riccati equations (ESREs) on infinite time horizon. Approximated by a sequence of BSDEs on finite time horizon, we prove that the two systems of ESREs admit nonnegative solutions. Eventually we solve the stochastic LQ problem with the help of the ESREs and some approximation techniques. The uniqueness of solutions for the ESREs is deduced by a verification argument.
